Linearized Semiclassical Initial Value Representation Research Articles

Semiclassical evaluation of expectation values.

Semiclassical mechanics allows for a description of quantum systems which preserves their phase information, and thus interference effects, while using only the system's classical dynamics as an input. In particular one of the strengths of a semiclassical description is to present a coherent picture which (to negligible higher-order ℏ corrections) is independent of the particular canonical coordinates used. However, this coherence relies heavily on the use of the stationary phase approximation. It turns out, however, that in some important cases, a brutal application of stationary phase approximation washes out all interference, and thus quantum, effects. In this paper, we address this issue in detail in one of its simplest instantiations, namely the evaluation of the time evolution of the expectation value of an operator. We explain why it is necessary to include contributions which are not in the neighborhood of stationary points and provide new semiclassical expressions for the evolution of the expectation values. The efficiency of our approach is based on the fact that we treat analytically all the integrals that can be performed within the stationary phase approximation, implying that the remaining integrals are simple integrals, in the sense that the integrand has no significant variations on the quantum scale (and thus they are very easy to perform numerically). This to be contrasted with other approaches such as the ones based on initial value representation, popular in chemical and molecular physics, which avoid a root search for the classical dynamics, but at the cost of performing numerically integrals whose evaluation requires a sampling on the quantum scale, and which are therefore not well designed to address the deep semiclassical regime. Along the way, we get a deeper understanding of the origin of these interference effects and an intuitive geometric picture associated with them.

On computing spectral densities from classical, semiclassical, and quantum simulations.

The Caldeira-Leggett model provides a compact characterization of a thermal environment in terms of a spectral density function, which has led to a variety of numerically exact quantum methods for reduced density matrix propagation. Since spectral densities are often computed from classical molecular dynamics simulations, we investigate in this paper whether quantum effects should be accounted for in the calculations. Therefore, we reformulate the recently developed Fourier method for spectral density calculations from semiclassical simulations which approximately allow for quantum effects. We propose two possible protocols based on either correlation functions or expectation values. These protocols are tested on a generic Calderra-Leggett model for the linearized semiclassical initial-value representation (LSC-IVR), the thawed Gaussian wave packet dynamics (TGWD), and hybrid schemes combining the two with the more accurate Herman-Kluk formula. Surprisingly, spectral densities from the LSC-IVR method, which treats the dynamics completely classically, are found to be extremely accurate, even in the quantum regime, where this method does not give a correct description of the correlation functions and expectation values. In contrast, the TGWD method turns out as too inaccurate for spectral density calculations, and the hybrid schemes perform well only if the system is close to the classical regime. This implies that, if the bath has a Caldeira-Leggett form, spectral densities are insensitive to quantum effects and any effort to approximately account for them rather leads to errors. Hence, in this case, spectral densities can be computed from classical simulations and used in a reduced quantum simulation as well.

On the identity of the identity operator in nonadiabatic linearized semiclassical dynamics.

Simulating the nonadiabatic dynamics of condensed-phase systems continues to pose a significant challenge for quantum dynamics methods. Approaches based on sampling classical trajectories within the mapping formalism, such as the linearized semiclassical initial value representation (LSC-IVR), can be used to approximate quantum correlation functions in dissipative environments. Such semiclassical methods however commonly fail in quantitatively predicting the electronic-state populations in the long-time limit. Here we present a suggestion to minimize this difficulty by splitting the problem into two parts, one of which involves the identity and treating this operator by quantum-mechanical principles rather than with classical approximations. This strategy is applied to numerical simulations of spin-boson model systems, showing its potential to drastically improve the performance of LSC-IVR and related methods with no change in the equations of motion or the algorithm in general, but rather by simply using different functional forms of the observables.

On the accuracy of the LSC-IVR approach for excitation energy transfer in molecular aggregates.

We investigate the applicability of the linearized semiclassical initial value representation (LSC-IVR) method to excitation energy transfer (EET) problems in molecular aggregates by simulating the EET dynamics of a dimer model in a wide range of parameter regime and comparing the results to those obtained from a numerically exact method. It is found that the LSC-IVR approach yields accurate population relaxation rates and decoherence rates in a broad parameter regime. However, the classical approximation imposed by the LSC-IVR method does not satisfy the detailed balance condition, generally leading to incorrect equilibrium populations. Based on this observation, we propose a post-processing algorithm to solve the long time equilibrium problem and demonstrate that this long-time correction method successfully removed the deviations from exact results for the LSC-IVR method in all of the regimes studied in this work. Finally, we apply the LSC-IVR method to simulate EET dynamics in the photosynthetic Fenna-Matthews-Olson complex system, demonstrating that the LSC-IVR method with long-time correction provides excellent description of coherent EET dynamics in this typical photosynthetic pigment-protein complex.

Recent advances in the linearized semiclassical initial value representation/classical Wigner model for the thermal correlation function

This article focuses on most recent advances in the linearized semiclassical initial value representation (LSC‐IVR)/classical Wigner model that includes quantum effects with classical trajectories and recovers exact thermal correlation functions (of even nonlinear operators, that is, nonlinear functions of position or momentum operators) in the classical, high temperature, and harmonic limits. Two methods for implementing the LSC‐IVR/classical Wigner which are in principle feasible to be combined with general force fields or even ab initio electronic structure methods have been reviewed. One is the local Gaussian approximation with the imaginary time path integral approach, the other is the quantum thermal bath method. The article emphasizes on the theory and the algorithms for implementation, while it also covers recent applications and limitations of the LSC‐IVR/classical Wigner. © 2015 Wiley Periodicals, Inc.

Generalized dephasing relation for fidelity and application as an efficient propagator

The dephasing relation (DR), a linearization of semiclassical fidelity, is generalized to include the overlap of "off-diagonal" elements. The accuracy of the formulation is tested in integrable and chaotic systems and its scaling with dimensionality is studied in a Caldeira-Leggett model with many degrees of freedom. It is shown that the DR is often in very good agreement with numerically analytic quantum results and frequently outperforms an alternative semiclassical treatment. Most importantly, since there is no computationally expensive prefactor, and Monte Carlo Metropolis sampling is used to facilitate the calculation, the DR is found to scale remarkably well with increasing dimension. We further demonstrate that a propagator based on the DR can include more quantum coherence and outperform other popular linearized semiclassical methods, such as forward-backward semiclassical dynamics (FBSD) and the linearized semiclassical initial value representation (LSC-IVR).

Insights in quantum dynamical effects in the infrared spectroscopy of liquid water from a semiclassical study with an ab initio-based flexible and polarizable force field

The dynamical properties of liquid water play an important role in many processes in nature. In this paper, we focus on the infrared (IR) absorption spectrum of liquid water based on the linearized semiclassical initial value representation (LSC-IVR) with the local Gaussian approximation (LGA) [J. Liu and W. H. Miller, J. Chem. Phys. 131, 074113 (2009)] and an ab initio based, flexible, polarizable Thole-type model (TTM3-F) [G. S. Fanourgakis and S. S. Xantheas, J. Chem. Phys. 128, 074506 (2008)]. Although the LSC-IVR (LGA) gives the exact result for the isolated three-dimensional shifted harmonic stretching model, it yields a blueshifted peak position for the more realistic anharmonic stretching potential. By using the short-time information of the LSC-IVR correlation function; however, it is shown how one can obtain more accurate results for the position of the stretching peak. Due to the physical decay in the condensed phase system, the LSC-IVR (LGA) is a good and practical approximate quantum approach for the IR spectrum of liquid water. The present results offer valuable insight into future attempts to improve the accuracy of the TTM3-F potential or other ab initio-based models in reproducing the IR spectrum of liquid water.

Zero point energy leakage in condensed phase dynamics: An assessment of quantum simulation methods for liquid water

The approximate quantum mechanical ring polymer molecular dynamics (RPMD) and linearized semiclassical initial value representation (LSC-IVR) methods are compared and contrasted in a study of the dynamics of the flexible q-TIP4P/F water model at room temperature. For this water model, a RPMD simulation gives a diffusion coefficient that is only a few percent larger than the classical diffusion coefficient, whereas a LSC-IVR simulation gives a diffusion coefficient that is three times larger. We attribute this discrepancy to the unphysical leakage of initially quantized zero point energy (ZPE) from the intramolecular to the intermolecular modes of the liquid as the LSC-IVR simulation progresses. In spite of this problem, which is avoided by construction in RPMD, the LSC-IVR may still provide a useful approximation to certain short-time dynamical properties which are not so strongly affected by the ZPE leakage. We illustrate this with an application to the liquid water dipole absorption spectrum, for which the RPMD approximation breaks down at frequencies in the O-H stretching region owing to contamination from the internal modes of the ring polymer. The LSC-IVR does not suffer from this difficulty and it appears to provide quite a promising way to calculate condensed phase vibrational spectra.

Quantum dynamical effects in liquid water: A semiclassical study on the diffusion and the infrared absorption spectrum

The important role of liquid water in many areas of science from chemistry, physics, biology, geology to climate research, etc., has motivated numerous theoretical studies of its structure and dynamics. The significance of quantum effects on the properties of water, however, has not yet been fully resolved. In this paper we focus on quantum dynamical effects in liquid water based on the linearized semiclassical initial value representation (LSC-IVR) with a quantum version of the simple point charge/flexible (q-SPC/fw) model [Paesani et al., J. Chem. Phys. 125, 184507 (2006)] for the potential energy function. The infrared (IR) absorption spectrum and the translational diffusion constants have been obtained from the corresponding thermal correlation functions, and the effects of intermolecular and intramolecular correlations have been studied. The LSC-IVR simulation results are compared with those predicted by the centroid molecular dynamics (CMD) approach. Although the LSC-IVR and CMD results agree well for the broadband for hindered motions in liquid water, the intramolecular bending and O-H stretching peaks predicted by the LSC-IVR are blueshifted from those given by CMD; reasons for this are discussed. We also suggest that the broadband in the IR spectrum corresponding to restricted translation and libration gives more information than the diffusion constant on the nature of quantum effects on translational and rotational motions and should thus receive more attention in this regard.

Linearized semiclassical initial value time correlation functions with maximum entropy analytic continuation

The maximum entropy analytic continuation (MEAC) method is used to extend the range of accuracy of the linearized semiclassical initial value representation (LSC-IVR)/classical Wigner approximation for real time correlation functions. LSC-IVR provides a very effective "prior" for the MEAC procedure since it is very good for short times, exact for all time and temperature for harmonic potentials (even for correlation functions of nonlinear operators), and becomes exact in the classical high temperature limit. This combined MEAC+LSC/IVR approach is applied here to two highly nonlinear dynamical systems, a pure quartic potential in one dimensional and liquid para-hydrogen at two thermal state points (25 and 14 K under nearly zero external pressure). The former example shows the MEAC procedure to be a very significant enhancement of the LSC-IVR for correlation functions of both linear and nonlinear operators, and especially at low temperature where semiclassical approximations are least accurate. For liquid para-hydrogen, the LSC-IVR is seen already to be excellent at T=25 K, but the MEAC procedure produces a significant correction at the lower temperature (T=14 K). Comparisons are also made as to how the MEAC procedure is able to provide corrections for other trajectory-based dynamical approximations when used as priors.

On the calculation of vibrational energy relaxation rate constants from centroid molecular dynamics simulations

We explore the use of centroid molecular dynamics (CMD) for calculating vibrational energy relaxation (VER) rate constants of high-frequency molecular vibrations in the condensed phase. We employ our recently proposed linear-response-theory-based approach to VER [Q. Shi and E. Geva, J. Chem. Phys. 118, 7562 (2003)], to obtain a new expression for the VER rate constant in terms of a correlation function that can be directly obtained from CMD simulations. We show that the new expression reduces to a centroid Landau-Teller-type formula in the golden-rule regime. Unlike previously proposed CMD-based approaches to VER, the new formula does not involve additional assumptions beyond the inherent CMD approximation. The new formula has the same form as the classical Landau–Teller formula, and quantum effects enter it in two ways: (1) The initial sampling and subsequent dynamics are governed by the centroid potential, rather than the classical potential; (2) The classical force is replaced by the corresponding centroid symbol. The application of the new method is reported for three model systems: (1) A vibrational mode coupled to a harmonic bath, with the coupling exponential in the bath coordinates; (2) A diatomic molecule coupled to a short linear chain of Helium atoms; (3) A “breathing sphere” diatomic molecule in a two-dimensional monoatomic Lennard-Jones liquid. It is confirmed that CMD is able to capture the main features of the force–force correlation function rather well, in both time and frequency domains. However, we also find that CMD is unable to accurately predict the high-frequency tail of the quantum-mechanical power spectrum of this correlation function, which limits its usefulness for calculating VER rate constants of high-frequency molecular vibrations. The predictions of CMD are compared with those obtained via the linearized-semiclassical initial-value-representation (LSC-IVR) method, which does yield accurate predictions of high-frequency VER rate constants. The reasons underlying these observations are discussed in terms of the similarities and differences between these two approaches.

A relationship between semiclassical and centroid correlation functions

A general relationship is established between semiclassical and centroid-based methods for calculating real-time quantum-mechanical correlation functions. It is first shown that the linearized semiclassical initial-value-representation (LSC-IVR) approximation can be obtained via direct linearization of the forward-backward action in the exact path integral expression for the correlation function. A Kubo-transformed two-time correlation function, with the position operator as one of the two operators, is then cast in terms of a carefully crafted exact path integral expression. Linearization of the corresponding forward–backward action, supplemented by the assumption that the dynamics of the centroid is decoupled from that of the higher normal modes, is then shown to lead to the centroid correlation function.

Rate constants and kinetic isotope effects for Cl+CH 4→ClH+CH 3: a comparison between LSC-IVR and statistical theories

Rate constants for Cl+CH 4→ClH+CH 3 (forward and reverse reactions) were computed by the linearized semiclassical-initial value representation (LSC-IVR) method. Also, the H/D kinetic isotope effects for the forward reaction were calculated. Two statistical theories, specifically, conventional transition state theory (TST) and canonical unified statistical (CUS) theory, were applied for the sake of comparison. The agreement between LSC-IVR, CUS and experiment is reasonably good over the range of temperatures studied here (400–2500 K).

Semiclassical study of electronically nonadiabatic dynamics in the condensed-phase: Spin-boson problem with Debye spectral density

The linearized semiclassical initial value representation (LSC-IVR) [H. Wang, X. Sun and W. H. Miller, J. Chem. Phys. 108, 9726 (1998)] is used to study the nonadiabatic dynamics of the spin-boson problem, a system of two electronic states linearly coupled to an infinite bath of harmonic oscillators. The spectral density of the bath is chosen to be of the Debye form, which is often used to model the solution environment of a charge transfer reaction. The simulation provides a rather complete understanding of the electronically nonadiabatic dynamics in a broad parameter space, including coherent to incoherent transitions along all three axes (the T-axis, the η-axis, and the ωc-axis) in the complete phase diagram and the determination of rate constants in several physically interesting regimes. Approximate analytic theories are used to compare with the simulation results, and good agreement is found in the appropriate physical limits.